cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A154597 a(n) = 15*a(n-1) + a(n-2) with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 15, 226, 3405, 51301, 772920, 11645101, 175449435, 2643386626, 39826248825, 600037119001, 9040383033840, 136205782626601, 2052127122432855, 30918112619119426, 465823816409224245, 7018275358757483101, 105739954197771470760, 1593117588325329544501
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009

Keywords

Comments

Limit_{n -> infinity} a(n)/a(n-1) = (15 + sqrt(229))/2. - Klaus Brockhaus, Oct 07 2009
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 15's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n - 1 on alphabet {0,1,...,15} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Apr 30 2023: (Start)
Also called the 15-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 15 kinds of squares available. (End)

Links

Crossrefs

Row n=15 of A073133, A172236 and A352361 and column k=15 of A157103.
First bisection is A098247.
Cf. A166125 (decimal expansion of sqrt(229)), A166126 (decimal expansion of (15 + sqrt(229))/2).
Cf. also A041427, A090301, A098245.
Sequences with g.f. 1/(1-k*x-x^2) or x/(1-k*x-x^2): A000045 (k=1), A000129 (k=2), A006190 (k=3), A001076 (k=4), A052918 (k=5), A005668 (k=6), A054413 (k=7), A041025 (k=8), A099371 (k=9), A041041 (k=10), A049666 (k=11), A041061 (k=12), A140455 (k=13), A041085 (k=14), this sequence (k=15), A041113 (k=16), A178765 (k=17), A041145 (k=18), A243399 (k=19), A041181 (k=20).

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-229); S:=[ ((15+r)^n-(15-r)^n)/(2^n*r): n in [1..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009
  • Magma
    [n le 2 select n-1 else 15*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 20 2024
  • Mathematica
    LinearRecurrence[{15,1}, {0,1}, 31] (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
CoefficientList[Series[x/(1-15*x-x^2), {x,0,50}], x] (* G. C. Greubel, Apr 16 2017 *)
  • PARI
    my(x='x+O('x^50)); concat([0], Vec(x/(1-15*x-x^2))) \\ G. C. Greubel, Apr 16 2017
  • SageMath
    def A154597(n): return (-i)^(n-1)*chebyshev_U(n-1, 15*i/2)
[A154597(n) for n in range(31)] # G. C. Greubel, Sep 20 2024

Formula

G.f.: x/(1 - 15*x - x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 07 2009
a(n) = ((15 + sqrt(229))^n - (15 - sqrt(229))^n)/(2^n*sqrt(229)).
From Johannes W. Meijer, Jun 12 2010: (Start)
Limit_{k -> infinity} a(n+k)/a(k) = (A090301(n) + a(n)*sqrt(229))/2.
Limit_{n -> infinity} A090301(n)/a(n) = sqrt(229).
a(2n+1) = 15*A098245(n-1).
a(3n+1) = A041427(5n), a(3n+2) = A041427(5n+3), a(3n+3) = 2*A041427(5n+4). (End)
E.g.f.: (2/sqrt(229))*exp(15*x/2)*sinh(sqrt(229)*x/2). - G. C. Greubel, Sep 20 2024

Extensions

Extended beyond a(7) by Klaus Brockhaus and Philippe Deléham, Jan 12 2009
Name from Philippe Deléham, Jan 12 2009
Edited by Klaus Brockhaus, Oct 07 2009
Missing a(0) added by Jianing Song, Jan 29 2019

A041427 Denominators of continued fraction convergents to sqrt(229).

Original entry on oeis.org

1, 7, 8, 15, 113, 3405, 23948, 27353, 51301, 386460, 11645101, 81902167, 93547268, 175449435, 1321693313, 39826248825, 280105435088, 319931683913, 600037119001, 4520191516920, 136205782626601, 957960669903127, 1094166452529728, 2052127122432855
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The a(n) terms of this sequence can be constructed with the terms of sequence A154597. For the terms of the periodical sequence of the continued fraction for sqrt(229) see A040213. We observe that its period is five. - Johannes W. Meijer, Jun 12 2010

Links

Crossrefs

Cf. A041426, A166125, A040213, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.

Programs

  • Magma
    I:=[1,7,8,15,113,3405,23948,27353,51301,386460]; [n le 10 select I[n] else 3420*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 17 2013
  • Mathematica
    Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[229], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[229], 30]] (* Vincenzo Librandi, Dec 17 2013 *)
LinearRecurrence[{0,0,0,0,3420,0,0,0,0,1},{1,7,8,15,113,3405,23948,27353,51301,386460},30] (* Harvey P. Dale, Oct 14 2020 *)

Formula

a(5*n) = A154597(3*n+1), a(5*n+1) = (A154597(3*n+2) - A154597(3*n+1))/2, a(5*n+2) = (A154597(3*n+2) + A154597(3*n+1))/2, a(5*n+3) = A154597(3*n+2) and a(5*n+4) = A154597(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: -(x^8 -7*x^7 +8*x^6 -15*x^5 +113*x^4 +15*x^3 +8*x^2 +7*x +1) / (x^10 +3420*x^5 -1). - Colin Barker, Nov 12 2013
a(n) = 3420*a(n-5) + a(n-10) for n>9. - Vincenzo Librandi, Dec 17 2013

A166126 Decimal expansion of 1/(imaginary part of (15+2*I)^(1/2))^2.

Original entry on oeis.org

1, 5, 0, 6, 6, 3, 7, 2, 9, 7, 5, 2, 1, 0, 7, 7, 7, 9, 6, 3, 5, 9, 5, 9, 3, 1, 0, 2, 4, 6, 7, 0, 5, 3, 2, 6, 0, 5, 8, 6, 2, 4, 3, 7, 7, 4, 1, 9, 2, 5, 9, 8, 5, 0, 9, 1, 1, 4, 3, 4, 5, 1, 4, 9, 6, 4, 9, 1, 4, 0, 5, 5, 5, 1, 7, 5, 8, 5, 0, 8, 3, 8, 6, 2, 3, 1, 1, 4, 7, 6, 6, 8, 8, 7, 4, 3, 7, 9, 8, 5, 5, 5, 6, 9, 3
Offset: 2

Views

Author

Klaus Brockhaus, Oct 07 2009

Keywords

Comments

lim_{n -> infinity} b(n)/b(n-1) = 1/imag((15+2*I)^(1/2))^2 for b = A154597.
Contribution from Klaus Brockhaus, May 28 2010: (Start)
Also decimal expansion of (15+sqrt(229))/2.
Continued fraction expansion of (15+sqrt(229))/2 is A010854. (End)

Examples

			1/imag((15+2*I)^(1/2))^2 = 15.06637297521077796....

Crossrefs

A154597, A166125 (decimal expansion of sqrt(229)).
Cf. A010854 (all 15's sequence). [From Klaus Brockhaus, May 28 2010]

A178233 Decimal expansion of (11+sqrt(229))/18.

Original entry on oeis.org

1, 4, 5, 1, 8, 1, 9, 2, 1, 9, 4, 6, 7, 8, 6, 4, 2, 1, 8, 1, 7, 7, 3, 2, 5, 6, 6, 9, 4, 0, 7, 8, 3, 6, 9, 5, 6, 2, 0, 6, 9, 3, 7, 5, 2, 6, 8, 8, 0, 6, 6, 5, 0, 1, 0, 1, 2, 7, 0, 5, 0, 1, 6, 6, 2, 7, 6, 8, 2, 2, 8, 3, 9, 0, 8, 4, 2, 7, 8, 7, 0, 9, 5, 8, 1, 2, 3, 8, 6, 2, 9, 8, 7, 4, 9, 3, 1, 0, 9, 5, 0, 6, 3, 2, 5
Offset: 1

Views

Author

Klaus Brockhaus, May 23 2010

Keywords

Comments

Continued fraction expansion of (11+sqrt(229))/18 is A069705.

Examples

			(11+sqrt(229))/18 = 1.45181921946786421817...

Crossrefs

Cf. A166125 (decimal expansion of sqrt(229)), A069705 (repeat 1, 2, 4).

A178236 Decimal expansion of (7+sqrt(229))/18.

Original entry on oeis.org

1, 2, 2, 9, 5, 9, 6, 9, 9, 7, 2, 4, 5, 6, 4, 1, 9, 9, 5, 9, 5, 5, 1, 0, 3, 4, 4, 7, 1, 8, 5, 6, 1, 4, 7, 3, 3, 9, 8, 4, 7, 1, 5, 3, 0, 4, 6, 5, 8, 4, 4, 2, 7, 8, 7, 9, 0, 4, 8, 2, 7, 9, 4, 4, 0, 5, 4, 6, 0, 0, 6, 1, 6, 8, 6, 2, 0, 5, 6, 4, 8, 7, 3, 5, 9, 0, 1, 6, 4, 0, 7, 6, 5, 2, 7, 0, 8, 8, 7, 2, 8, 4, 1, 0, 3
Offset: 1

Views

Author

Klaus Brockhaus, May 23 2010

Keywords

Comments

Continued fraction expansion of (7+sqrt(229))/18 is A153727.

Examples

			(7+sqrt(229))/18 = 1.22959699724564199595...

Crossrefs

Cf. A166125 (decimal expansion of sqrt(229)), A153727 (repeat 1, 4, 2).

Programs

  • Mathematica
    RealDigits[(7+Sqrt[229])/18,10,120][[1]] (* Harvey P. Dale, Aug 23 2019 *)
Showing 1-5 of 5 results.

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